Vagueness

There is wide agreement that a term is vague to the extent that it has
borderline cases. This makes the notion of a borderline case crucial
in accounts of vagueness. I shall concentrate on an historical
characterization of borderline cases that most commentators would
accept. Vagueness will then be contrasted with ambiguity and
generality. This will clarify the nature of the philosophical
challenge posed by vagueness. I will then discuss some rival theories
of vagueness with an emphasis on many-valued logic, supervaluationism
and contextualism. I will conclude with the issue of whether all
vagueness is linguistic.

If you cut one head off of a two headed man, have you decapitated
him? What is the maximum height of a short man? When does a fertilized
egg develop into a person?

These questions are impossible to answer because they involve absolute
borderline cases. In the vast majority of cases, the unknowability of
a borderline statement is only relative to a given means of settling
the issue (Sorensen 2001, chapter 1). For instance, a boy may count
as a borderline case of ‘obese’ because people cannot tell
whether he is obese just by looking at him. A curious mother could try
to settle the matter by calculating her son's body mass index. The
formula is to divide his weight (in kilograms) by the square of his
height (in meters). If the value exceeds 30, this test counts him as
obese. The calculation will itself leave some borderline cases. The
mother could then use a weight-for-height chart. These charts are not
entirely decisive because they do not reflect the ratio of fat to
muscle, whether the child has large bones, and so on. The boy will
only count as an absolute borderline case of ‘obese’ if no
possible method of inquiry could settle whether he is obese. When we
reach this stage, we start to suspect that our uncertainty is due to
the concept of obesity rather than to our limited means of testing for
obesity.

Absolute borderline cases are targeted by Charles Sander Peirce's
entry for ‘vague’ in the 1902 Dictionary of Philosophy
and Psychology:

A proposition is vague when there are possible states of
things concerning which it is intrinsically uncertain whether,
had they been contemplated by the speaker, he would have regarded them
as excluded or allowed by the proposition. By intrinsically uncertain
we mean not uncertain in consequence of any ignorance of the
interpreter, but because the speaker's habits of language were
indeterminate. (Peirce 1902, 748)

In the case of relative borderline cases, the question is clear but
our means for answering it are incomplete. In the case of absolute
borderline cases, there is incompleteness in the question itself.

When a term is applied to one of its absolute borderline cases the
result is a statement that resists all attempts to settle whether it is
true or false. No amount of conceptual analysis or empirical inquiry
can settle whether removing one head from a two headed man counts as
decapitating him. We could give the appearance of settling the matter
by stipulating that ‘decapitate’ means ‘remove a
head’ (as opposed to ‘make headless’ or ‘remove
the head’ or ‘remove the most important head’). But
that would amount to changing the topic to an issue that merely sounds
the same as decapitation.

Vagueness is standardly defined as the possession of borderline
cases. For example, ‘tall’ is vague because a man who is
1.8 meters in height is neither clearly tall nor clearly non-tall. No
amount of conceptual analysis or empirical investigation can settle
whether a 1.8 meter man is tall. Borderline cases are inquiry
resistant. Indeed, the inquiry resistance typically recurses. For in
addition to the unclarity of the borderline case, there is normally
unclarity as to where the unclarity begins. In other words
‘borderline case’ has borderline cases. This higher order
vagueness seems to show that ‘vague’ is vague.

Higher order vagueness appears to condemn us to draw a sharp line somewhere. If the line is not drawn between the true and the false, then it will be between the true and the intermediate state. Introducing further intermediates just delays the inevitable.

The only way out is to reject a presupposition of the problem. Accordingly, several philosophers characterize higher order vagueness as an illusion (Wright 2010). They deny that there is an open-ended iteration of borderline status. Speakers do not go around talking about borderline borderline cases and borderline borderline borderline cases and so forth (Raffman 2005, 23).

Defenders of higher order vagueness say that ordinary speakers avoid iterating
‘borderline’ for the same reason they avoid iterating ‘million’
or ‘know’.
The iterations are confusing but perfectly meaningful.
‘Borderline’
behaves just like a vague predicate. For instance,
‘borderline’
can be embedded in a sorites argument. Defenders of higher order vagueness have also tried to clinch the case with particular specimens such as borderline hermaphrodites (reasoning that these individuals are borderline borderline males) (Sorensen 2010).

‘Tall’ is relative. A 1.8 meter pygmy is tall for a
pygmy but a 1.8 meter Masai is not tall for a Masai. Although
relativization disambiguates, it does not eliminate borderline cases.
There are shorter pygmies who are borderline tall for a pygmy and
taller Masai who are borderline tall for a Masai. The direct bearers of
vagueness are a word's full disambiguations such as ‘tall for an
eighteenth century French man’. Words are only vague indirectly,
by virtue of having a sense that is vague. In contrast, an ambiguous
word bears its ambiguity directly—simply in virtue of having
multiple meanings.

This contrast between vagueness and ambiguity is obscured by the
fact that most words are both vague and ambiguous. ‘Child’
is ambiguous between ‘offspring’ and ‘immature
offspring’. The latter reading of ‘child’ is vague
because there are borderline cases of immature offspring. The contrast
is further complicated by the fact that most words are also general.
For instance, ‘child’ covers both boys and girls.

Ambiguity and vagueness also contrast with respect to the speaker's
discretion. If a word is ambiguous, the speaker can resolve the
ambiguity without departing from literal usage. For instance, he can
declare that he meant ‘child’ to express the concept of an
immature offspring. If a word is vague, the speaker cannot resolve the
borderline case. For instance, the speaker cannot make
‘child’ literally mean anyone under eighteen just by
intending it. That concept is not, as it were, on the menu
corresponding to ‘child’. He would be understood as taking
a special liberty with the term to suit a special purpose. This departure from ordinary usage would
relieve him of the obligation to defend the sharp cut-off.

When the movie director Alfred Hitchcock mused ‘All actors are
children’ he was taking liberties with clear negative cases of
‘child’ rather than its borderline cases. The aptness of
his generalization is not judged by its literal truth-value (because
it is obviously untrue). Likewise, we do not judge precisifications of
borderline cases by their truth-values (because they are obviously not
ascertainable as true or false). We instead judge precisifications by
their simplicity, conservativeness, and fruitfulness. A
precisification that draws the line across the borderline cases
conserves more paradigm usage than one that draws the line across
clear cases. But conservatism is just one desideratum among many.
Sometimes the best balance is achieved at the cost of turning former
positive cases into negative cases.

Once we shift from literal to figurative usage, we gain fictive
control over our entire vocabulary—not just vague words. When a
travel agent says ‘France is a hexagon’, we do not infer
that she has committed the geometrical error of classifying France as
a six sided polygon. We instead interpret the travel agent as speaking
figuratively, as meaning that France is shaped like a
hexagon. Similarly, when the travel agent says ‘Reno is the
biggest little city’, we do not interpret her as overlooking the
vagueness of ‘little city’. Just as she uses the obvious
falsehood of ‘France is a hexagon’ to signal a metaphor,
she uses the obvious indeterminacy of ‘Reno is the biggest
little city’ to signal hyperbole.

Given that speakers lack any literal discretion over vague terms, we
ought not to chide them for indecisiveness. Where there is no decision
to be made, there is no scope for vice.

Speakers would have literal discretion if statements applying a
predicate to its borderline cases were just permissible variations in
linguistic usage. For instance, Crispin Wright and Stewart Shapiro say
a competent speaker can faultlessly classify the borderline case as a
positive instance while another competent speaker can faultlessly
classify the case as a negative instance.

For the sake of comparison, consider discretion between alternative
spellings. Professor Letterman uses ‘judgment’ instead of
‘judgement’ because he wants to promote the principle that
a silent E signals a long vowel. He still has fond memories of Tom
Lehrer's 1971 children's song “Silent E”:

Who can turn a can into a cane?
Who can turn a pan into a pane?
It's not too hard to see,
It's Silent E.

Who can turn a cub into a cube?
Who can turn a tub into a tube?
It's elementary
For Silent E.

Professor Letterman disapproves of those who add the misleading E but
concedes that ‘judgement’ is a permissible spelling; he
does not penalize his students for misspelling when they make their
hard-hearted choice of ‘judgement’. Indeed, like other
professors, he scolds students if they fail to stick with the same
spelling throughout the composition. Choose but stick to your choice!

Professor Letterman's assertion ‘The word for my favorite mental
act is spelled j-u-d-g-m-e-n-t’ is robust with respect to the
news that it is also spelled j-u-d-g-e-m-e-n-t. He would continue to
assert it. He can conjoin the original assertion with information
about the alternative: ‘The word for my favorite mental act is
spelled j-u-d-g-m-e-n-t and is also spelled
j-u-d-g-e-m-e-n-t’. In contrast, Professor Letterman's assertion
that ‘Martha is a woman’ is not robust with respect to the
news that Martha is a borderline case of ‘woman’ (say,
Letterman learns Martha is younger than she looks). The new
information would lead Letterman to retract his assertion in favor of
a hedged remark such as ‘Martha might be a woman and Martha
might not be a woman’. Professor Letterman's loss of confidence
is hard to explain if the information about her borderline status were
simply news of a different but permissible way of describing
her. Discoveries of notational variants do not warrant changes in
former beliefs.

News of borderline status has an evidential character. Loss of clarity
brings loss of warrant. If you do not lower your confidence, you are
open to the charge of dogmatism. To concede that Martha is a
borderline case of ‘woman’ is to concede that you do not
know that she is a woman. That is why debates can be dissolved by
showing that the dispute is over a borderline case. The debaters
should be agnostic if they are dealing with a borderline case. They do
not have a license to form beliefs beyond their evidence.

News of an alternative sense is like news of an alternative spelling;
there is no evidential impact (except for meta-linguistic beliefs
about the nature of words). Your assertion that ‘All bachelors
are men’ is robust with respect to the news that
‘bachelor’ has an alternative sense in which it means a
male seal. Assertions are not robust with respect to news of hidden
generality. If a South African girl says ‘No elephant can be
domesticated’ but is then taught that there is another species
of elephant indigenous to Asia, then she will lose some confidence;
maybe Asian elephants can be domesticated. News of hidden
generality has evidential impact. When it comes to robustness,
vagueness resembles generality more than vagueness resembles
ambiguity.

Mathematical terms such as ‘prime number’ show that a
term can be general without being vague. A term can also be vague
without being general. Borderline cases of analytically empty
predicates illustrate this possibility.

Generality is obviously useful. Often, lessons about a particular
F can be projected to other Fs in virtue of their
common F-ness. When a girl learns that her cat has a
nictating membrane that protects its eyes, she rightly expects her
neighbor's cat also has a nictating membrane. Generality saves labor.
When the girl says that she wants a toy rather than clothes, she
narrows the range of acceptable gifts without going through the trouble
of specifying a particular gift. The girl also balances values: a gift
should be intrinsically desired and yet also be a surprise. If
uncertain about which channel is the weather channel, she can hedge by
describing the channel as ‘forty-something’. There is an
inverse relationship between the contentfulness of a proposition and
its probability: the more specific a claim, the less likely it is to be
true. By gauging generality, we can make sensible trade-offs between
truth and detail.

‘Vague’ has a sense which is synonymous with abnormal
generality. This precipitates many equivocal explanations of vagueness.
For instance, many commentators say that vagueness exists because broad
categories ease the task of classification. If I can describe your
sweater as red, then I do not need to figure out whether it is scarlet.
This freedom to use wide intervals obviously helps us to learn, teach,
communicate, and remember. But so what? The problem is to explain the
existence of borderline cases. Are they present because vagueness
serves a function? Or are borderline cases side-effects of ordinary
conversation—like echoes?

Every natural language is both vague and ambiguous. However, both
features seem eliminable. Indeed, both are eliminated in miniature
languages such as checkers notation, computer programming languages,
and mathematical descriptions. Moreover, it seems that both vagueness
and ambiguity ought to be minimized. ‘Vague’ and
‘ambiguous’ are pejorative terms. And they deserve their
bad reputations. Think of all the automotive misery that has been
prefaced by

Driver: Do I turn left?
Passenger: Right.

English can be lethal. Philosophers have long motivated appeals for
an ideal language by pointing out how ambiguity creates the menace of
equivocation:

No child should work.
Every person is a child of someone.
Therefore, no one should work.

Happily, we know how to criticize and correct all equivocations.
Indeed, every natural language is self-disambiguating in the sense that
each has all the resources needed to uniquely specify any reading one
desires. Ambiguity is often the cause but rarely the object of
philosophical rumination.

Vagueness, in contrast, precipitates a profound problem: the sorites
paradox. For instance,

Base step: A one day year old human being is a child.

Induction step: If an n day old human being is a child,
then that human being is also a child when it is n + 1 days
old.

Conclusion: Therefore, a 36,500 day old human being is a child.

The conclusion is false because a 100 year old man is clearly a
non-child. Since the base step of the argument is also plainly true and
the argument is valid by mathematical induction, we seem to have no
choice but to reject the second premise.

George Boolos (1991) observes that we have an autonomous case
against the induction step. In addition to implying plausible
conditionals such as ‘If a 1 day old human being is a child, then
that human being is also a child when it is 2 days old’, the
induction step also implies ludicrous conditionals such as ‘If a
1 day old human being is a child, then that human being is also a child
when it is 36,500 days old’. For some reason, we tend to overlook
these easy counterexamples to the induction step.

With Boolos' helping hand, we have driven a second stake into the
heart of the sorites paradox. Yet the paradox seems far from dead.
The negation of the second premise classically implies a sharp
threshold for childhood. For it implies the existential generalization
that there is a number n such that an n day old
human being is a child but is no longer a child one day later.

Epistemicists accept this astonishing consequence. They think
vagueness is a form of ignorance. Timothy Williamson (1994) traces the
ignorance of the threshold for childhood to
“margin for error”
principles. If one knows that an n day old human being is a
child, then that human being must also be a child when n + 1
days old. Otherwise, one is right by luck. Given that there is a
threshold, we would be ignorant of its location.

Several critics focus on attitudes weaker than knowledge. According to Nicholas Smith (2008, 182) we cannot guess that the threshold for baldness is the 400th hair. Hartry Field (2010, 203) denies that a rational man can fear that he has just passed the threshold into being old. Hope, speculation, and wonder do not require evidence but they do require understanding. So it is revealing that these attitudes have trouble getting a purchase on the threshold of oldness (or any other vague predicate). A simple explanation is that bare linguistic competence gives us knowledge that are no such thresholds. This accounts for the comical air of the epistemicist. Just as there is no conceptual room to worry that there is a natural number between sixty and sixty one, there is no conceptual room to worry that one has passed the threshold of oldness between one's sixtieth and sixty first birthday.

An old epistemicist might reply:
My piecemeal confidence that a given number is not the threshold for oldness does not agglomerate into collective confidence that there is no such number. If I bet against each number being the threshold, then I must have placed a losing bet somewhere. For if I won each bet then there was no opportunity for me to make the transition to oldness. My bookie could have made “Dutch book” against me. He would have been entitled to payment without having to identify which bet I lost. Since probabilities may be extracted from hypothetical betting behavior, I must actually assign some small (normally negligible) probability to hypotheses identifying particular thresholds. So must you.

Stephen Schiffer (2003, 204) denies that classical probability calculations apply in vague contexts. Suppose Ned is borderline old and borderline bald. According to Schiffer
we should be just as confident in the conjunction
‘Ned is old and bald’
as in either conjunct. Adding conjuncts does not reduce confidence because we have a “vague partial belief”rather than the standard belief assumed by mathematicians developing probability theory. Schiffer offers a calculus for this vagueness-related propositional attitude. He crafts the rules for vague partial belief to provide a psychological solution to the sorites paradox.

The project is complicated by the fact that vague partial beliefs interact with precise beliefs (MacFarlane 2010). Consider a statement that has a mixture of vague and precise conjuncts:
‘Ned is old and bald and has an even number of hairs’. Adding the extra precise conjunct should diminish confidence. Schiffer also needs to accommodate the fact that some speakers are undecided about whether the nature of the uncertainty involves vagueness. Even an idealized speaker may unsure because there is vagueness about the borders between vagueness related uncertainty and other sorts of uncertainty.

Other commentators grant that it is logically possible that vague predicates have thresholds. They just think it would be a miracle:
“It is logically possible that the words on this page will come to life and sort my socks. But I know enough about words to dismiss this as a serious possibility. So I am right to boggle at the possibility that
our rough and ready terms such as ‘red’ could so sensitively classify objects.”
Epistemicists counter that this
bafflement rests on an over-estimate of the role of stipulation in
meaning. Epistemicists say much meaning is acquired passively by default rather
than actively by decision.

Most philosophers doubt whether precise analytical tools fit vague
arguments. H. G. Wells was amongst the first to suggest that we must
moderate the application of logic:

Every species is vague, every term goes cloudy at its
edges, and so in my way of thinking, relentless logic is only another
name for stupidity—for a sort of intellectual pigheadedness. If
you push a philosophical or metaphysical enquiry through a series of
valid syllogisms—never committing any generally recognized
fallacy—you nevertheless leave behind you at each step a
certain rubbing and marginal loss of objective truth and you get
deflections that are difficult to trace, at each phase in the process.
Every species waggles about in its definition, every tool is a little
loose in its handle, every scale has its individual.—First
and Last Things (1908)

Many more believe that the problem is with logic itself rather than the
manner in which it is applied. They favor solving the sorites paradox
by replacing standard logic with an earthier deviant logic.

There is a desperately wide range of opinions as to how the revision
of logic should be executed. Every form of deviant logic has been
applied in the hope of resolving the sorites paradox.

An early favorite was many-valued logic. On this approach,
borderline statements are assigned truth-values that lie between full
truth and full falsehood. Some logicians favor three truth-values, others
prefer four or five. The most popular approach is to use an infinite number of truth-values
represented by the real numbers between 0 (for full falsehood) and 1 (for
full truth). This infinite spectrum of truth-values might be service for a continuous sorites argument involving
‘small real number’
(Weber and Colyvan 2010).

Critics object that this proliferation of truth-values exacerbates the
over-precision of classical logic. Instead of having just one
artificially sharp line between the truth and the false, the
many-valued logician has infinitely many sharp lines such as that
between statements with a truth of of .323483925 and those with a
higher truth-value. In Mark Sainsbury's words, “… you do not
improve a bad idea by iterating it.” (1996, 255)

A proponent of an infinite valued logic might reply to Sainsbury with
an analogy. It is a bad idea to model a circle with a straight
line. Using two lines is not much better, nor is there is much
improvement using a three sided polygon (a triangle). But as we add
more straight lines to the polygon (square, pentagon, hexagon, and so
on) we make progress—by iterating the bad idea of modeling a circle
with straight lines.

Indeed, it would be tempting to triumphantly conclude ‘The
circle has been modeled as an infinitely sided polygon’. This
victory declaration would itself need clarification. Has
the circle been revealed to be an infinitely sided polygon? Have
curved lines been replaced by straight lines? Have curved lines (and
hence circles) been proven to not exist? A model can succeed without
it being clear what has been achieved.

But it is premature to dwell on the simile ‘Precision is to
vagueness as straightness is to curvature’. The many-valued
logician must first vindicate the analogy by providing details about
how to calculate the truth-values of vague statements from the
truth-values of their component statements.

Proponents of many-valued logic approach this obligation with great
industry. Precise new rules are introduced to calculate the truth
value of compound statements that contain statements with intermediate
truth-values. For instance, the revised rule for conjunctions is to
assign the conjunction the same truth-value as the conjunct with the
lowest truth-value.

These rules are designed to yield all standard theorems when all the
truth values are 1 and 0. In this sense, classical logic is a limiting
case of many-valued logic. Classical logic is agreed to work fine in
the area for which it was designed—mathematics.

Most theorems of standard logic break down when intermediate
truth-values are involved. (An irregular minority, such as ‘If
P, then P’, survive.) Even the classical
contradiction ‘Bozo is bald and it is not the case that he is
bald’ receives a truth-value of .5 when ‘Bozo is
bald’ has a truth-value of .5. Many-valued logicians note that
the error they are imputing to classical logic is often so small that
classical logic can still be fruitfully applied. But they insist that
the sorites paradox illustrates how tiny errors can accumulate into a
big error.

Critics of the many-valued approach complain that it botches
phenomena such as hedging. If I regard you as a borderline case of
‘tall man’, I cannot sincerely assert that you are tall and
I cannot sincerely assert that you are of average height. But I can
assert the hedged claim ‘You are tall or of average
height’. The many-valued rule for disjunction is to assign the
whole statement the truth-value of its highest disjunct. Normally, the
added disjunct in a hedged claim is not more plausible than the other
disjuncts. Thus it cannot increase the degree of truth.
Disappointingly, the proponent of many-valued logic cannot trace the
increase of assertibility to an increase the degree of truth.

Epistemicists explain the rise in assertibility by the increasing
probability of truth. Since the addition of disjuncts can raise
probability indefinitely, the epistemicists correctly predict that we
can hedge our way to full assertibility. However, epistemicists do
not have a monopoly on this prediction.

According to supervaluationists, borderline statements lack a
truth-value. This neatly explains why it is universally impossible to
know the truth-value of a borderline statement. Supervaluationists
offer details about the nature of absolute borderline cases. Simple
sentences about borderline cases lack a truth-value. Compounds of these
statements can have a truth-value if they come out true regardless of
how the statement is precisified. For instance, ‘Either Mr. Stoop
is tall or it is not the case that Mr. Stoop is tall’ is true
because it comes out true under all ways of sharpening
‘tall’. Thus the method of supervaluations allows one to
retain all the theorems of standard logic while admitting “truth-value
gaps”.

One may wonder whether this striking result is a genuine convergence
with standard logic. Is the supervaluationist characterizing vague
statements as propositions? Or is he merely pointing out that certain
non-propositions have a structure isomorphic to logical theorems? (Some
electrical circuits are isomorphic to tautologies but this does not make
the circuits tautologies.) Kit Fine (1975, 282), and especially David
Lewis (1982), characterize vagueness as hyper-ambiguity. Instead of there
being one vague concept, there are many precise concepts that closely
resemble each other. ‘Child’ can mean a human being at most
one day old or mean a human being at most two days old or mean a human
being at most three days old …. Thus the logic of vagueness is a
logic for equivocators. Lewis' idea is that ambiguous statements are
true when they come out true under all disambiguations. But logicians
normally require that a statement be disambiguated before
logic is applied.
The mere fact that an ambiguous statement comes out true under all its
disambiguations does not show that the statement itself is true.
Sentences which are actually disambiguated may have
truth-values. But the best that can be said of those that merely
could be disambiguated is that they would have had a
truth-value had they been disambiguated (Tye 1989).

Supervaluationism will converge with classical logic only if each word
of the supervaluated sentence is uniformly interpreted. For instance,
‘Either a carbon copy of Teddy Roosevelt's signature is an
autograph or it is not the case that a carbon copy of Teddy
Roosevelt's signature is an autograph’ comes out true only if
‘autograph’ is interpreted the same way in both disjuncts.
Vague sentences resist mixed interpretations. However, mixed
interpretations are permissible for ambiguous sentences. As Lewis
himself notes in a criticism of relevance logic, ‘Scrooge walked
along the bank on his way to the bank’ can receive a mixed
disambiguation. When exterminators offer ‘non-toxic ant
poison’, we charitably switch relativizations within the noun
phrase: the substance is safe for human beings but deadly for
ants.

Even if one agrees that supervaluationism converges with classical
logic about theoremhood, they clearly differ in other respects.
Supervaluationism requires rejection of inference rules such as
contraposition, conditional proof and reductio ad absurdum (Williamson
1994, 151–152). In the eyes of the supervaluationist, a demonstration
that a statement is not true does not guarantee that the statement is
false.

The supervaluationist is also under pressure to reject semantic
principles which are intimately associated with the application of
logical laws. According to Alfred Tarski's Convention T, a statement
‘S’ is true if and only if S. In other
words, truth is disquotational. Supervaluationists say that being
supertrue (being true under all precisifications) suffices for being
true. But given Convention T, supertruth would then be disquotational.
Since the supervaluationists accept the principle of excluded middle,
they would be forced to say ‘P’ is supertrue or
‘Not P’ is supertrue (even if
‘P’ applies a predicate to a borderline case).
This would imply that either ‘P’ is true or
‘Not P’ is true. (Williamson 1994, 162–163) And
that would be a fatal loss of truth-value gaps for
supervaluationism.

There is a final concern about the “ontological honesty” of the
supervaluationist's existential quantifier. As part of his solution to
the sorites paradox, the supervaluationist will assert ‘There is
a human being who was a child when n days old but not when
n + 1 days old’. For this statement comes out true under
all admissible precisifications of ‘child’. However, when
pressed the supervaluationist will add an unofficial clarification:
“Oh, of course I do not mean that there really is a sharp threshold for
childhood.”

After the clarification, some wonder how supervaluationism differs
from drastic metaphysical skepticism. In his nihilist days, Peter Unger
(1979) admitted that it is useful to talk as if there are
children. But he insisted that strictly speaking, vague terms such as
‘child’ cannot apply to anything. Unger was free to use
supervaluationism as a theory to explain our ordinary discourse about
children. (Unger instead used other resources to explain how we
fruitfully apply empty predicates.) But once the dust had cleared and
the precise rubble came into focus, Unger had to conclude that there
are no children.

Officially, the supervaluationist rejects the induction step of the
sorites argument. Unofficially, he seems to instead reject the
base step of the sorites argument.

Supervaluationism is also haunted by a logical analogy. Whereas the supervaluationist analyzes borderline cases in terms of truth-value gaps the dialetheist analyzes them in terms of truth-value gluts. A glut is a proposition that is both true and false. The rule for assigning gluts is the mirror image of the rule for assigning gaps:
A statement is true exactly if it comes out true on at least one precisification. The statement is false just if it comes out false on at least one precisification. So if the statement comes out true under one precisification and false under another precisification, the statement is both true and false.

To avoid triviality, the dialetheist must adopt a logic that stops two contradictory statements from jointly implying everything. The resulting “subvaluationism” is an ingenious dual of supervaluationism. Viewed formally, there seems no more reason to prefer one departure from classical logic rather than the other. Since Western philosophers abominate contradiction, parity with dialetheism would diminish the great popularity of supervaluationism.

A Machiavellian epistemicist will welcome this battle between the gaps and the gluts. He roots for the weaker side. Although he does not want the subvaluationist to win, the Machiavellian epistemicist does want the subvaluationist to achieve mutual annihilation with his supervaluationist doppelgänger. His political calculation is: Gaps + Gluts = Bivalence.

Pablo Corberos (2011) has argued that subvaluationism provides a
better treatment of higher order vagueness than supervaluationism. But
for the most part, the subvaluationists (and their frenemies) have
merely claimed subvaluationism to be at least as attractive as
supervaluationism (Hyde and Colyvan 2008). This modest ambition seems
prudent. After all, truth-value gaps have far more independent support
from the history of philosophy. Prior to the explosive growth of
vagueness research after 1975, ordinary language philosophers amassed
a panoramic battery of analyses suggesting that gaps are involved in
presupposition, reference failure, fiction, future contingent
propositions, performatives, and so on and so on. Supervaluationism
rigorously consolidated these appeals to ordinary language.

Dialetheists characterize intolerance for contradiction as a shallow
phenomenon, restricted to a twentieth Western academic milieu (maybe
even now being eclipsed by the rise of China). Experimental
philosophers have challenged the old appeals to ordinary language with
empirical results suggesting that glutty talk is as readily stimulated
by borderline cases as gappy talk (Alxatib and Pelletier 2011, Ripley
2011).

Just as contextualism in epistemology runs orthogonal to the familiar
divisions amongst epistemologists (foundationalism, reliabilism,
coherentism, etc.), there are contextualists of every persuasion
amongst vagueness theorists. They develop an analogy between the
sorites paradox and indexical sophistries such as:

1. Base step: The horizon is more than 1 meter away.

2. Induction step: If the horizon is more than n meters
away, then it is more than n + 1 meters away.

3. Conclusion: The horizon is more than a billion meters
away.

The horizon is where the earth meets the sky and is certainly less
than a billion meters away. (The circumference of the earth is only
forty million meters.) Yet when you travel toward the horizon to
specify the n at which the induction step fails, your trip is
as futile as the pursuit of the rainbow. You cannot reach the horizon
because it shifts with your location.

All contextualists accuse the sorites monger of equivocating. In one
sense, the meaning of ‘child’ is uniform; the
context-invariant rule for using the term (its
“character”) is constant. However, the set of things to
which the term applies (its “content”) shifts with the
context. In this respect, vague words resemble indexical terms such
as: I, you, here, now, today, tomorrow. When a debtor tells his
creditor on Monday ‘I will pay you back tomorrow’ and then
repeats the sentence on Tuesday, there is a sense in which he has said
the same thing (the character is the same) and a sense in which he has
said something different (the content has shifted because
‘tomorrow’ now picks out Wednesday).

According to the contextualists, the rules governing the shifts
prohibit us from interpreting any instance of the induction step as
having a true antecedent and a false consequent. The very process of
trying to refute the induction step changes the context so that the
instance will not come out false. Indeed, contextualists typically
emphasize that each instance is true and so compel our assent. Direct
attacks on the induction step can never be successful. One is put in
mind of Seneca's admonition to his student Nero: “However many you put to
death, you will never kill your successor.”

How strictly are we to take the comparison between vague words and
indexical terms? Scott Soames (2002, 445) answers that all vague words
literally are indexical.

This straightforward response is open to the objection that the
sorites monger could stabilize reference. When the sorites monger
relativizes ‘horizon’ to the northeast corner of the
Empire State Building's observation deck, he seems to generate a
genuine sorites paradox that exploits the vagueness of
‘horizon’ (not its indexicality).

All natural languages have stabilizing pronouns, ellipsis, and other
anaphoric devices. For instance, in ‘Jack is tired now and Jill
is too’, the ‘too’ forces a uniform reading of
‘tired’. Jason Stanley suggests that the sorites monger
employ the premise:

If that1 is a child then that2 is
too, and if that2 is too, then that3 is too, and if
that3 is too, then that4 is too, … and then
thati is too.

Each ‘thatn’ refers to the
nth element in a sequence of worsening examples of
‘child’. The meaning of ‘child’ is not
shifting because the first occurrence of the term governs all the
subsequent clauses (thanks to ‘too’). If vague terms were
literally indexical, the sorites monger would have a strong reply. If
vague terms only resemble indexicals, then the contextualist needs to
develop the analogy in a way that circumvents Stanley's counsel to the
sorites monger.

There is certainly no shortage of more guarded contextualists. Hans
Kamp, the founder of contextualism, maintained that the extension of
vague words orbits the speaker's store of conversational commitments.
In a more psychological vein, Diana Raffman says changes in context
trigger gestalt shifts between look-alike categories.

Stewart Shapiro integrates Kamp's ideas with Friedrich Waismann's
concept of open texture. Shapiro thinks speakers have discretion over
borderline cases because they are judgment dependent. They come out
true in virtue of the speaker judging them to be true. Given that the
audience does not resist, borderline cases of ‘child’ can
be correctly described as children. The audience recognizes that
other competent speakers could describe the borderline case
differently. As Waismann lyricizes “Every description stretches,
as it were, into a horizon of open possibilities: However far I go, I
shall always carry this horizon with me.” (1968, 122)

American pragmaticism colors Delia Graff Fara's contextualism. Consider
dandelion farms. Why would someone grow weeds? The answer is that
‘weed’ is relative to interests. Dandelions are unwanted
by lawn caretakers but are wanted by farmers for food, wine, and
medical uses. Delia Fara thinks this interest relativity extends to
all vague words. For instance, ‘child’ means a degree of
immaturity that is significant to the speaker. Since the interests of
the speaker shifts over time, there is an opportunity for a shift in
the extension of ‘child’. Graff is reluctant to describe
herself as a contextualist because the context only has an indirect
effect on the extension via the changes it makes to the speaker's
interest.

The first challenge for the contextualist is to find enough shiftiness
to block every sorites argument. Since vagueness reaches into every
syntactic category, critics complain that contextualism is
exceeds the level of ambiguity countenanced by linguistics and
psycholinguists. Analogy: hydrologists agree there is much hidden
water but all hydrologists are scientifically committed to denying
Thales' claim that all is water.

Another concern is that some sorites arguments involve predicates that do not
give us an opportunity to equivocate. Consider a sorites with a base step that starts from a number too large for us to think about. (There are infinitely many of these.) Or consider an inductive predicate that is too complex for us to reason with. One example is obtained by iterating ‘mother of’ a thousand
times (Sorensen 2001, 33). This predicate could be embedded in a mind numbing
sorites that would never generate context shifts.

Other unthinkable sorites argument use predicates that can only be
grasped by individuals in other possible worlds or by creatures with
different types of minds than ours. More fancifully, there could be a
vague predicate, such as Saul Kripke's “killer yellow”, that instantly kills anyone who thinks about it. The basic problem is that contextualism is a psychologistic theory of the sorites. Arguments can exist without being propounded.

Supervaluationists encourage the view that all vagueness is a matter
of linguistic indecision: the reason why there are borderline cases is
that we have not bothered to make up our minds. Many
supervaluationists maintain that this indecision is
functional. Instead of committing ourselves prematurely, we can fill
in meanings as we go along in light of new information and
interests. This conjecture is promising for the highly stipulative
enterprise of promulgating and enforcing laws (Endicott 2000). Judges
frequently seem to exercise and control discretion by means of vague
language.

Discretion through gap-filling pleases those who regard adjudication as a creative process. It alarms those think we should be judged by laws rather than men.
The doctrine of discretion through indeterminacy has also been questioned on grounds that the source of the discretion is the generality of the legal terms rather than their
vagueness (Poscher 2012).

Supervaluationists emphasize the distinction between words and objects.
Objects themselves do not seem to be the sort of thing that can be
general, ambiguous, or vague (Eklund 2011). From this perspective, Georg Hegel appears to be committing a category mistake when he characterizes clouds as
vague. Although we sometimes speak of clouds being ambiguous or even being general to a region, this does not entitle us to infer that there is metaphysical ambiguity or metaphysical generality.

Supervaluationists are here incorporating an orthodoxy dating back to Bertrand Russell's seminal article “Vagueness” (1923). This consensus was re-affirmed by Michael Dummett (1975) and regularly re-avowed by subsequent commentators.

In 1978 Gareth Evans focused opposition to vague objects with a short proof modeled after Saul Kripke's attack on contingent identity. If there is a vague object, then some statement of the form ‘a = b’ must be vague (where each of the flanking singular terms precisely designates that object). For the vagueness is allegedly due to the object rather than its representation. But any statement of form
‘a = a’ is definitely true. Consequently, a has the property of being definitely identical to a. Since a = b, then b must also have the property of being definitely identical to a. Therefore ‘a = b’ must be definitely true!

Evans agrees that there are vague identity statements in which one of the flanking terms is vague (just as Kripke agree that there are contingent identity statements when one of the flanking terms is a flaccid designator). But then the vagueness is due to language, not the world.

Despite Evans' impressive assault, there was a renewal of interest in vague objects in the 1980s. As a precedent for this revival, Peter van Inwagen (1990, 283)
recalls that in the 1960s, there was a consensus that all necessity is
linguistic. Most philosophers now take the possibility of essential
properties seriously.

Some of the reasons are technical. Problems with Kripke's refutation of contingent identity tend to have structural parallels that affect Evans' proof. Evans also relies on inferences that deviant logicians challenge (Parsons 2000).

In the absence of a decisive reductio ad absurdum , many logicians feel their role to be the liberal one of articulating the logical space for vague objects. There should be “Vague objects for those who want them” (Cowles and White 1991). Logic should be ontologically neutral.

Some non-enemies of vague objects also have an ambition to consolidate various species of indeterminacy (Barnes and Williams 2011). Talk of indeterminacy is found in quantum mechanics, analyses of the open future, fictional incompleteness, and the continuum hypothesis. Perhaps vagueness is just one face of indeterminacy.

This panoramic vision contrasts with the continuing resolution of many to tether vagueness to the sorites paradox (Eklund 2011). They fear that the clarity achieved by semantic ascent will be muddied by metaphysics.

But maybe the mud is already on the mountain top. Trenton Merricks (2001) claims that standard characterizations of linguistic vagueness rely on metaphysical vagueness. If ‘Bozo is bald’ lacks a truth-value because there
is no fact to make the statement true, then the shortage appears to be
ontological.

The view that vagueness is always linguistic has been attacked from
other directions. Consider the vagueness of maps (Varzi 2001). The vagueness is pictorial rather than discursive. So one cannot conclude that vagueness is linguistic merely from the premise that vagueness is representational.

Or consider vague instrumental music such as Claude Debussy's “The Clouds”.
Music has syntax but too little semantics to qualify as language. There is a little diffuse reference through devices such as musical quotation, leitmotifs, and homages. These referential devices are not precise. Therefore, some music is vague (Sorensen 2010). The strength and significance of this argument depends on the relationship between music and language. Under the musilanguage hypothesis, language and music branched off from a common “musilanguage” with language specializing in semantics and music specializing in the expression of emotion. This scenario makes it plausible that purely instrumental music could have remnants of semantic meaning.

Mental imagery also seems vague. When rising suddenly
after a prolonged crouch, I “see stars before my eyes”. I can tell
there are more than ten of these hallucinated lights but I cannot tell
how many. Is this indeterminacy in thought to be reduced to
indeterminacy in language? Why not vice versa? Language is an
outgrowth of human psychology. Thus it seems natural to view language
as merely an accessible intermediate bearer of vagueness.

–––, 1993, “Many, but almost one”, in Ontology,
Causality, and Mind: Essays on the Philosophy of D.M. Armstrong,
Keith Campbell, John Bacon, and Lloyd Reinhardt (eds.), Cambridge:
Cambridge University Press.